Re: Independent vs uncorrelated

Ben wrote:


R.V. are independent if the joint PDF of X and Y= (PDF of X) * ( PDF
of Y). This is the direct result of the fact that if X and Y are
independent than conditioning does not change the PDF.

Yes thats what I meant by:

P(X,Y,Z,.....) = P(X)P(Y)P(Z)..... <-----> independence ------->
uncorrelated

However, the relation btw


R.V are uncorrelated if Expectation {XY} =E{x}E{Y}.

and

spherical symmetry of joint pdf P(X,Y,Z,....) ----> uncorrelated (rxy,
ryz, rxz,..... all are dirac functions)

is not so clear!!! Any thoughts.


I was actually a bit sloppy. Spherical symmetry and zero mean imply
uncorrelatedness, but spherical symmetry by itself does not. Further
spherical symmetry implies equal variances of the RVs. For now lets assume
that the elements of random vector x = (x0, x1, ..., xn)^T have variances
equal to 1 and that they have zero mean.

If the elements of the x are not correlated then the covariance matrix is:

E{x*x^T} = I

and any arbitrary rotation of the vector x results in uncorrelated random
vector y:

y = A*x, where A is orthogonal (orthonormal basis)
E{y*y^T} = A*E{x*x^T}*A^T = A*I*A^T = I

That is, uncorrelatedness is invariant under rotation. If the elements of
the x are correlated with each other then

E{x*x^T} = B

and

y = A*x, where A is orthogonal (orthonormal basis)
E{y*y^T} = A*B*A^T = C =/= B.

Thus if the elemenst of the vector x are correlated, the correlation is
variant under rotation.

Spherically symmetric distribution is invariant under rotation. The
correlation between the elements of spherically symmetrically distributed
random vector do not change under rotation. Thus, if the distribution is
spherically symmetric the corresponding random variables have to be
uncorrelated.

Please correct me if I was inaccurate or made a mistake.

--
Jani Huhtanen
Tampere University of Technology, Pori
.

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